Cyclically five-connected cubic graphs

A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdividing two distinct edges of $G$ and joining the two new vertices by an edge, or $G'$ is obtained from $G$ by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from $G$ in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of $H$ we are able to eliminate the second construction. We also prove versions of both of these results when $G$ is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case $G'$ is required to be almost cyclically 5-connected and to have fewer circuits of length four than $G$. In particular, if $G$ has at most one circuit of length four, then $G'$ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs $G'$ are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To appear in J. Combin. Theory Ser.

Declarative Specification

Deriving formal specifications from informal requirements is extremely difficult since one has to overcome the conceptual gap between an application domain and the domain of formal specification methods. To reduce this gap we introduce application-specific specification languages, i.e., graphical and textual notations that can be unambiguously mapped to formal specifications in a logic language. We describe a number of realised approaches based on this idea, and evaluate them with respect to their domain specificity vs. generalit

K-6 minors in large 6-connected graphs

Jorgensen conjectured that every 6-connected graph with no K-6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs. (C) 2017 Published by Elsevier Inc

Bespoke controllers and their impact on game feel

This project explores the role and effect of bespoke game controllers on game feel and accessibility. The project aims to develop a range of custom controllers and associated games, designed to tightly couple the user interaction with the game execution. Player analytics will be collected to quantify, explore and improve the user experience.Game feel encompasses a wide range of definitions including intuitive controls, physical interaction with virtual objects, and the appeal of playing. Game feel is subjective and therefore challenging to quantify, but something every player instinctively recognises. There is a growing niche of games utilising bespoke hardware to deliver a unique player experience, such as games played with Unicorn horns or by shouting, rather than the traditional controller. The project seeks to push the boundaries of games and explore how game design, user experience and accessibility are affected when the limitation of traditional input systems is removed. The project has developed four bespoke controllers, utilising creative technologies and physical sensors. The controllers are based on inputs using force sensitive sensors, flex sensors, distance sensor and orientation sensors. Prototype games have been developed for each controller designed to exploit the controllers unique form of user interaction. The next steps are to evaluate the player experience and explore how the various controllers are utilised using a data-driven approach.The project proposes that the bespoke controller could widen engagement and improve the user experience by delivering a more intuitive interface and improving game feel. The project will provide insights and design guidelines for both of the development of the custom game controllers and how they can be linked to certain game mechanics in the associated games